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Collaborative beamforming has been widely used in wireless sensor networks to improve the directivity of signals in long-distance transmission. The performance of collaborative beamforming has been well analyzed for the case without phase offset in the literature. However, there always exists the phase ambiguity caused by carrier phase jitter between the transmitter and receiver nodes in a practical system. Although the effects of imperfect phase have been studied for Uniform node distribution and Tikhonov phase noise distribution, the performance analysis of collaborative beamforming with arbitrary node distributions and any phase offset is still an open issue. This paper proposes a unified method to evaluate the performance of collaborative beamforming in the case of phase noise. Since the non-parametric kernel method is used to build the probability density functions of node and phase offset distributions, the proposed nonparametric approach can provide accurate performance analysis for various node and phase noise distributions. The simulation results verify the proposed method.

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We present the triangular lattice pattern with the invisible Kagome lattice in the dielectric barrier discharge system with two water electrodes for the first time. The spatio-temporal dynamics studied by an intensified charge-coupled device (ICCD) show that the complex pattern is composed of four different sublattices: Kagome lattice, halos, and big and small spots (B and S). The invisible Kagome lattice is similar to the dark discharge because of its spatio-temporal randomness. Based on the novel discharges (B and S) occurring at the falling edge of applied voltage, the dynamics of surface charges is investigated and demonstrated with a high speed video camera and photomultiplier tubes. Furthermore, the formation of restricted random structures is considered to be the effect of previous discharges which rearrange the surface charges.